On the Arithmetic of Power Monoids and Sumsets in Cyclic Groups

Abstract

Let $H$ be a multiplicatively written monoid with identity $1_H$ (in particular, a group). We denote by ${\mathcal{P}}_{\mathrm{fin},\times }(H)$ the monoid obtained by endowing the collection of all finite subsets of $H$ containing a unit with the operation of setwise multiplication $(X,Y)\mapsto \{xy:x\in X,y\in Y\}$; and study fundamental features of the arithmetic of this and related structures, with a focus on the submonoid, ${\mathcal{P}}_{\text{fin},1}(H)$, of ${\mathcal{P}}_{\text{fin},\times }(H)$ consisting of all finite subsets $X$ of $H$ with $1_H\in X$. Among others, we prove that ${\mathcal{P}}_{\text{fin},1}(H)$ is atomic (i.e., each non-unit is a product of irreducibles) iff $1_H\ne x^2\ne x$ for every $x\in H\setminus \{1_H\}$. Then we obtain that ${\mathcal{P}}_{\text{fin},1}(H)$ is BF (i.e., it is atomic and every element has factorizations of bounded length) iff $H$ is torsion-free; and show how to transfer these conclusions to ${\mathcal{P}}_{\text{fin},\times }(H)$. Next, we introduce "minimal factorizations" to account for the fact that monoids may have non-trivial idempotents, in which case standard definitions from Factorization Theory degenerate. Accordingly, we obtain conditions for ${\mathcal{P}}_{\text{fin},\times }(H)$ to be BmF (meaning that each non-unit has minimal factorizations of bounded length); and for ${\mathcal{P}}_{\text{fin},1}(H)$ to be BmF, HmF (i.e., a BmF-monoid where all the minimal factorizations of a given element have the same length), or minimally factorial (i.e., a BmF-monoid where each element has an essentially unique minimal factorization). Finally, we prove how to realize certain intervals as sets of minimal lengths in ${\mathcal{P}}_{\text{fin},1}(H)$. Many proofs come down to considering sumset decompositions in cyclic groups, so giving rise to an intriguing interplay with Arithmetic Combinatorics.

23 pp., 1 figure (on p. 4). Fixed minor details and added Sect. 2.4 and Remarks 4.2 and 4.6. To appear in Pacific Journal of Mathematics

References in corpus (3)

• Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics
• Factorizations in bounded hereditary Noetherian prime rings
• Structural properties of subadditive families with applications to factorization theory

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• On the arithmetic of power monoids
• Semigroups of ideals and isomorphism problems
• Factorization in monoids and rings
• Torsion groups and the Bienvenu--Geroldinger conjecture
• Factorization Theory in Commutative Monoids
• On the automorphisms of the power semigroups of a numerical semigroup